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Operator Method for Nonperturbative Calculation of the Thermodynamic Values in Quantum Statistics

И. Д. Феранчук, A. A. Ivanov

Published Sep 21, 2003• 0 citations
OpenAlex

Abstract

Operator method and cumulant expansion are used for nonperturbative calculation of the partition function and the free energy in quantum statistics. It is shown for some model systems that the zeroth approximation of the method proposed interpolates the thermodynamics values with rather good accuracy in the entire range of both the Hamiltonian parameters and temperature. The regular procedure for calculation of the corrections to the zeroth approximation is also considered.

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Renormalization and Statistical Mechanics in Many-Particle Systems. II. Statistical Perturbation Method

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A general nonperturbation problem, described by the Hamiltonian $\mathcal{H}$, is considered. Approximate creation operators ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ are defined as operators which satisfy the Hamiltonian commutator equations $[\mathcal{H},{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}]={\ensuremath{\omega}}_{i}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}+{{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, where ${\ensuremath{\omega}}_{i}$ are the creation energies, and the remainder operators ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ are small in the sense that statistical averages involving the ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ are small. A zeroth approximation is to neglect the ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$. In this case, a straight-forward derivation provides general relations between statistical averages such as $〈{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\Omega}〉$ and $〈{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\Omega}\ifmmode\pm\else\textpm\fi{}\ensuremath{\Omega}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}〉$, where $\ensuremath{\Omega}$ is any operator. The usual boson and fermion occupation numbers are a trivial result of these zeroth-order relations. Corrections to the general zeroth-order relations arise from the ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$; such corrections are considered in the spirit of perturbation theory. An explicit first-order correction is obtained for the averages $〈{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\theta}}_{i}〉$, and this correction is related to a self-consistent energy-renormalization procedure. The present method is similar to the method of thermodynamic Green's functions in that the derivations given here do not require evaluation of any state vectors or of the partition function. The general results are applied to the Heisenberg ferromagnet problem. At low temperatures, the zeroth-order relations give the Bloch spin-wave results, and an approximate evaluation of the first-order contribution gives the leading term in Dyson's ${T}^{4}$ correction to the spontaneous magnetization. For arbitrary temperatures, the zeroth-order relations give the Tyablikov equations for $〈{S}^{z}〉$, and Callen's results are obtained from the first-order corrections. These examples illustrate the simplicity of calculating statistical averages of functions of the ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, ${\ensuremath{\theta}}_{i}$ operators by the present method.

Physical Review•Dec 2, 1966•21 citations
Operator method and cumulant expansion are used for nonperturbative calculation of the partition function and the free energy in quantum statistics. It is shown for some model systems that the zeroth approximation of the method proposed interpolates the thermodynamics values with rather good accuracy in the entire range of both the Hamiltonian parameters and temperature. The regular procedure for calculation of the corrections to the zeroth approximation is also considered.
Nonperturbative method for description of quantum systems - the operator method (OM) \nand the conception of the uniformly suitable estimation (USE) are considered for the series of real physical systems. It is shown that the OM zeroth-order approximation permits one to find the analytical approximation for eigenfunctions and eigenvalues with high accuracy \nwithin the entire range of the Hamiltonian parameters and any quantum numbers. The \nOM subsequent approximations converge rapidly to the exact solutions of the Schrödinger equation. The generalization of OM for the quantum statistics is also developed.
We present in this paper a systematic nonperturbative cluster-cumulant method for deriving thermal averages of operators in quantum many-body systems. The method combines the advantages of the cumulant expansion scheme of thermodynamic perturbation theory, the approach of thermofield dynamics as a finite-temperature field theory, and the time-dependent coupled-cluster theory extended to ``imaginary time.'' We have generalized the concepts of cumulants in a nonperturbative manner and have posited on the statistical operator an exponential-like ansatz containing connected, size-extensive operators in the exponent. These latter cumulantlike operators have been termed ``cluster cumulants'' by us. For a compact treatment, we have derived an alternative thermal field theory in which a time-ordered product is expanded in terms of ``thermal normal products'' of operators and thermal contractions---leading to a ``thermal Wick's theorem.'' The thermal normal products are the finite-temperature analogs of the ordinary normal products and have zero thermal averages. Operators in these products commute (anticommute) under permutations for bosons (fermions). This thermal representation is shown to be unitarily related to the traditional thermofield dynamics formulation, but has the advantage of using only the physical variables. The imaginary-time evolution of the statistical operator is treated by our recently formulated time-dependent cluster-cumulant theory. The partition function is evaluated as an exponential of a connected quantity. As an illustrative example, we have computed the partition function of an anharmonic oscillator with equally weighted cubic and quartic perturbation for a wide range of coupling, extending to the strongly nonperturbative regime. We study the behavior of free energy in the low-temperature limit and verify numerically the validity of the Kohn-Luttinger theorem [Phys. Rev. 118, 41 (1960)] for this system. We also show that our formalism is a natural nonperturbative analog of the thermodynamic perturbative theory by showing that a perturbative solution of the thermal-cluster-cumulant equations generates a variation of the Bloch--Balian--de Dominicis theory.
A method is proposed for obtaining a systematic expansion of thermodynamic functions of spin systems with large spin S in powers of 1/ S . It uses the cumulant technique and a coherent-state representation of the partition function . The expansion of in terms of cumulants yields an effective classical Hamiltonian with temperature-dependent quantum corrections. For the Heisenberg quantum Hamiltonian, they have a non-Heisenberg form. The effective Hamiltonian can be solved by methods familiar for classical systems.
Abstract A systematic nonperturbative scheme for calculating the ground state energy is adapted for studying systems of strongly correlated electrons on a lattice. It includes a method for calculating the cumulants of the Hamiltonian and a method, using the t-expansion, for constructing successive approximations to the ground state energy by these cumulants. The scheme is applied to spinless fermion and Hubbard models, and a method is proposed to overcome the problems found in previous attempts to use this scheme to study the Hubbard model.
Schrödinger suggested that thermodynamical functions cannot be based on the gratuitous allegation that quantum-mechanical levels (typically the orthogonal eigenstates of the Hamiltonian operator) are the only allowed states for a quantum system [E. Schrödinger, Statistical Thermodynamics (Courier Dover, Mineola, 1967)]. Different authors have interpreted this statement by introducing density distributions on the space of quantum pure states with weights obtained as functions of the expectation value of the Hamiltonian of the system. In this work we focus on one of the best known of these distributions and prove that, when considered in composite quantum systems, it defines partition functions that do not factorize as products of partition functions of the noninteracting subsystems, even in the thermodynamical regime. This implies that it is not possible to define extensive thermodynamical magnitudes such as the free energy, the internal energy, or the thermodynamic entropy by using these models. Therefore, we conclude that this distribution inspired by Schrödinger's idea cannot be used to construct an appropriate quantum equilibrium thermodynamics.
A general nonperturbation problem, described by the Hamiltonian H\mathcal{H}H, is considered. Approximate creation operators ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ are defined as operators which satisfy the Hamiltonian commutator equations $[\mathcal{H},{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}]={\ensuremath{\omega}}_{i}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}+{{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, where ${\ensuremath{\omega}}_{i}$ are the creation energies, and the remainder operators ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ are small in the sense that statistical averages involving the ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ are small. A zeroth approximation is to neglect the ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$. In this case, a straight-forward derivation provides general relations between statistical averages such as $〈{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\Omega}〉$ and $〈{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\Omega}\ifmmode\pm\else\textpm\fi{}\ensuremath{\Omega}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}〉$, where $\ensuremath{\Omega}$ is any operator. The usual boson and fermion occupation numbers are a trivial result of these zeroth-order relations. Corrections to the general zeroth-order relations arise from the ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$; such corrections are considered in the spirit of perturbation theory. An explicit first-order correction is obtained for the averages $〈{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\theta}}_{i}〉$, and this correction is related to a self-consistent energy-renormalization procedure. The present method is similar to the method of thermodynamic Green's functions in that the derivations given here do not require evaluation of any state vectors or of the partition function. The general results are applied to the Heisenberg ferromagnet problem. At low temperatures, the zeroth-order relations give the Bloch spin-wave results, and an approximate evaluation of the first-order contribution gives the leading term in Dyson's T4{T}^{4}T4 correction to the spontaneous magnetization. For arbitrary temperatures, the zeroth-order relations give the Tyablikov equations for 〈Sz〉〈{S}^{z}〉〈Sz〉, and Callen's results are obtained from the first-order corrections. These examples illustrate the simplicity of calculating statistical averages of functions of the ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, ${\ensuremath{\theta}}_{i}$ operators by the present method.